An acoustic source in a fluid-filled borehole generates headwaves as well as relatively stronger borehole-guided modes. A standard sonic measurement system consists of placing a piezoelectric source and an array of hydrophone receivers inside a fluid-filled borehole. The piezoelectric source is configured in the form of either a monopole or a dipole source. The source bandwidth typically ranges from a 0.5 to 20 kHz. A monopole source generates primarily the lowest-order axi-symmetric mode, also referred to as the Stoneley mode, together with compressional and shear headwaves. In contrast, a dipole source primarily excites the lowest-order flexural borehole mode together with compressional and shear headwaves. The headwaves are caused by the coupling of the transmitted acoustic energy to plane waves in the formation that propagate along the borehole axis. An incident compressional wave in the borehole fluid produces critically refracted compressional waves in the formation. Those refracted along the borehole surface are known as compressional headwaves. The critical incidence angle θi=sin−1(Vf/Vc), where Vf is the compressional wave speed in the borehole fluid and Vc is the compressional wave speed in the formation. As the compressional headwave travels along the interface, it radiates energy back into the fluid that can be detected by hydrophone receivers placed in the fluid-filled borehole.
In fast formations, the shear headwave can be similarly excited by a compressional wave at the critical incidence angle θi=sin−1(Vf/Vs), where Vs is the shear wave speed in the formation. It is also worth noting that headwaves are excited only when the wavelength of the incident wave is smaller than the borehole diameter so that the boundary can be effectively treated as a planar interface. In a homogeneous and isotropic model of fast formations, compressional and shear headwaves can be generated by a monopole source placed in a fluid-filled borehole for determining the formation compressional and shear wave speeds. It is known that refracted shear headwaves cannot be detected in slow formations (where the shear wave velocity is less than the borehole-fluid compressional velocity) with receivers placed in the borehole fluid. In slow formations, formation shear velocities are obtained from the low-frequency asymptote of flexural dispersion. There are standard processing techniques for the estimation of formation shear velocities in either fast or slow formations from an array of recorded dipole waveforms.
Recorded waveforms at an array of hydrophone receivers placed in a fluid-filled borehole can be processed by a modified matrix pencil algorithm to isolate both non-dispersive and dispersive arrivals in the wavetrain. Both the lowest-order axi-symmetric Stoneley and flexural modes are dispersive, i.e., velocity changes as a function of frequency. It should be understood that three of the five independent anisotropic elastic constants can be obtained from the cross-dipole acoustic data and Stoneley data according to techniques known in the art. For example, in a vertical wellbore (e.g., parallel to X3-axis in FIG. 1), anisotropic elastic constants C44 and C55 can be estimated from the low-frequency asymptotes of cross-dipole acoustic data measured in the vertical wellbore, and anisotropic elastic constant C66 can be estimated from the Stoneley data measured in the vertical wellbore. In a horizontal wellbore (e.g., parallel to X1-axis in FIG. 1), anisotropic elastic constants C66 and C55 can be estimated from the low-frequency asymptotes of cross-dipole acoustic data measured in the horizontal wellbore, and anisotropic elastic constant C44 can be estimated from the Stoneley data measured in the horizontal wellbore. Further, the refracted compressional headwaves yield an estimate of anisotropic elastic constant C33 in a vertical wellbore and C11 in a horizontal wellbore.
Under these circumstances, it becomes necessary to combine sonic data from both a horizontal and deviated wellbores to estimate all five independent anisotropic elastic constants. This procedure assumes that both the deviated and horizontal (or vertical) wellbore trajectories are in the same homogeneous anisotropic formation. While such an assumption may be appropriate for constructing anisotropic velocity models for seismic (AVO) interpretation, it may lead to unreliable estimate of variations in near-wellbore stresses that influence hydraulic fracture propagation to aid in the productivity of shale-gas. It is, therefore, desirable to estimate all five independent elastic constants from sonic data acquired as a function of logging depth in a horizontal wellbore in a shale gas play that exhibits significant heterogeneity along the wellbore.